## Séminaire des thésards

### mercredi 20 février 2019 à 18:00 : Sophie Germain, salle 2015 Bâtiment Sophie Germain, av. de France métro Bibliothèque F. Mitterrand, Paris 13e

In 1987, Deligne proved a type of Riemann-Roch theorem, which aims to relate geometric and arithmetic properties of compact Riemann surfaces endowed with smooth hermitian metrics.
When trying to apply this result to the case of modular curves, we find that there is a crucial hypothesis that is not satisfied : the Poincaré metric does not behave nicely and has singularities at some points.
The purpose of this talk is to present a method, called analytic surgery, which we can use to avoid these singularities and get a variation of Deligne’s results. Some unexpected applications stem from these considerations, such as explicit values of some derivatives of Selberg zeta functions.

### mercredi 30 janvier 2019 à 18:00 : Jussieu, salle 15-16-413. 4 place Jussieu, 75005 Paris

Wikipedia gives the following definition for dynamical systems : "In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space". This talk will not respect Wikipedia’s definition, we will expand on it. Here, we will replace "time" (which often refers to $\mathbN$ or \$\mathbbR) by a Polish group and "a function (...) in a geometrical space" by a group action on a compact space.
Since a general Polish group can be hard to describe (specially since we will take them non-locally compact), we will restrict our groups as automorphism groups of countable structures. So there are definitely combinatorics involved (mostly graphs though) !
Trying to probe these groups, we are lead to the notions of amenability, unique ergodocity and extreme amenability. Those notions will be described in terms of probability measures and fixed points.